Skip to main content
SpringerLink
Log in

Dispersive traveling wave solutions to the space–time fractional equal-width dynamical equation and its applications

  • Published:
Optical and Quantum Electronics Aims and scope Submit manuscript

Abstract

In this article, some new traveling wave solutions to the space–time fractional equal-width equation are constructed with the help of the extended Fan sub-equation method. A simple transformation is introduced to convert the fractional order partial differential equation into an ordinary differential equation. As a result, the bright, dark, singular and combined wave solitons are observed for different values of two parameters. Moreover, the graphical representations are also depicted.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

References

  • Ablowitz, M.J., Clarkson, P.A.: Solitons, Nonlinear Evolution Equations and Inverse Scattering. Cambridge Univ. Press, New York (1991)

    Book  MATH  Google Scholar 

  • Ahmadian, S., Darvishi, M.T.: Fractional version of (1+1)-dimensional Biswas–Milovic equation and its solutions. Optik 127, 10135–10147 (2016)

    Article  ADS  Google Scholar 

  • Ahmadian, S., Darvishi, M.T.: New exact traveling wave solutions for space–time fractional (1+1)-dimensional SRLW equation. Optik 127, 10697–10704 (2016)

    Article  ADS  Google Scholar 

  • Annaby, M.H., Mansour, Z.S.: Q-Fractional Calculus and Equations, vol. 2056. Springer, New York (2012)

    MATH  Google Scholar 

  • Aziz, A., Khani, F., Darvishi, M.T.: Homotopy analysis method for variable thermal conductivity heat flux gage with edge contact resistance. Zeitschrift fuer Naturforschung A 65(10), 771–776 (2010)

    ADS  Google Scholar 

  • Bekir, A., Guner, O.: Exact solutions of nonlinear fractional differential equations by \((g^{\prime }/g)\)-expansion method. Chin. Phys. B 22, 1–6 (2013)

    ADS  Google Scholar 

  • Darvishi, M.T., Khani, F.: A series solution of the foam drainage equation. Comput. Math. Appl. 58(2), 360–368 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  • Darvishi, M.T., Khani, F., Hamedi-Nezhad, S., Ryu, S.-W.: New modification of the HPM for numerical solutions of the sine-Gordon and coupled sine-Gordon equations. Int. J. Comput. Math. 87(4), 908–919 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  • Dianchen, L., Seadawy, A.R., Arshad, M.: Applications of extended simple equation method on unstable nonlinear Schrdinger equations. Optik 140, 136–144 (2017)

    Article  ADS  Google Scholar 

  • Fan, H.L.: The classification of the single traveling wave solutions to the generalized equal width equation. Appl. Math. Comput. 219, 748–54 (2012)

    MathSciNet  MATH  Google Scholar 

  • Helal, M.A., Seadawy, A.R., Ibrahim, R.S.: Variational principle for Zakharov–Shabat equations in two-dimensions. Appl. Math. Comput. 219, 5635–5648 (2013)

    MathSciNet  MATH  Google Scholar 

  • Hirota, R.: Exact solutions of the Korteweg–de Vries equation for multiple collisions of solitons. Phys. Rev. Lett. 27, 1192–1194 (1971)

    Article  ADS  MATH  Google Scholar 

  • Jumarie, G.: Table of some basic fractional calculus formulae derived from a modified Riemann–Liouville derivative for non-differentiable functions. Appl. Math. Lett. 22, 378–385 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  • Kalim, U.T., Seadawy, A.R.: Bistable bright–dark solitary wave solutions of the (3+1)-dimensional breaking Soliton, Boussinesq equation with dual dispersion and modified KdV-KP equations and their applications. Results Phys. 7, 1143–1149 (2017)

    Article  ADS  Google Scholar 

  • Kalim, U.T., Younis, M.: Bright Dark and other optical solitons with second order spaotetimporal dispersion. Optik 142, 446–450 (2017)

    Article  ADS  Google Scholar 

  • Kalim, U.T., Younis, M., Rizvi, S.T.R.: Optical solitons in monomode fibers with higher order nonlinear Schr\(\ddot{o}\)dinger equation. Optik 154, 360–371 (2018)

    Article  ADS  Google Scholar 

  • Khalique, C.M., Adem, K.R.: Exact solutions of the (2+1)-dimensional Zakharov–Kuznetsov modified equal width equation using Lie group analysis. Math. Comput. Modell. 54, 184–189 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  • Khani, F., Hamedi-Nezhad, S., Darvishi, M.T., Ryu, S.W.: New solitary wave and periodic solutions of the foam drainage equation using the Exp-function method. Nonlinear Anal. Real World Appl. 10(3), 1904–1911 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  • Kichenassamy, S., Olver, P.: Existence and nonexistence of solitary wave solutions to higher-order model evolution equations. SIAM J. Math. Anal. 23(5), 1141–1166 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  • Korkmaz, A.: Exact solutions of space–time fractional EW and modified EW equations. Chaos Solitons Fractals 96, 132–138 (2017)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Li, Z.B., He, J.H.: Fractional complex transform for fractional differential equations. Math. Comput. Appl. 15, 970–973 (2010)

    MathSciNet  MATH  Google Scholar 

  • Mehdipoor, M., Neirameh, A.: New soliton solutions to the (3+1)-dimensional Jimbo–Miwa equation. Optik 126, 4718–4722 (2015)

    Article  ADS  Google Scholar 

  • Seadawy, A.R.: Stability analysis for two-dimensional ionacoustic waves in quantum plasmas. Phys. Plasmas 21(5) Article ID 052107 (2014)

  • Seadawy, A.R.: Stability analysis for Zakharov–Kuznetsov equation of weakly nonlinear ion-acoustic waves in a plasma. Comput. Math. Appl. 67(1), 172–180 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  • Seadawy, A.R.: Three-dimensional nonlinear modified Zakharov–Kuznetsov equation of ion-acoustic waves in a magnetized plasma. Comput. Math. Appl. 71, 201–212 (2016)

    Article  MathSciNet  Google Scholar 

  • Seadawy, A.R.: Stability analysis solutions for nonlinear three-dimensional modified Korteweg–de Vries–Zakharov–Kuznetsov equation in a magnetized electron–positron plasma. Phys. A 455, 44–51 (2016)

    Article  MathSciNet  Google Scholar 

  • Seadawy, A.R.: Solitary wave solutions of two-dimensional nonlinear Kadomtsev–Petviashvili dynamic equation in dust-acoustic plasmas. Pramana J. Phys. 89, 49 (2017a)

    Article  ADS  Google Scholar 

  • Seadawy, A.R.: Travelling wave solutions of a weakly nonlinear two-dimensional higher order Kadomtsev-Petviashvili dynamical equation for dispersive shallow water waves. Eur. Phys. J. Plus 29, 1–13 (2017b)

    Google Scholar 

  • Seadawy, A.R.: The generalized nonlinear higher order of KdV equations from the higher order nonlinear Schrödinger equation and its solutions. Optik 139, 31–43 (2017c)

    Article  ADS  Google Scholar 

  • Seadawy, A.R., Dianchen, L.: Bright and dark solitary wave soliton solutions for the generalized higher order nonlinear Schr\(\ddot{o}\)dinger equation and its stability. Results Phys. 7, 43–48 (2017)

    Article  Google Scholar 

  • Seadawy, A.R., El-Rashidy, K.: Traveling wave solutions for some coupled nonlinear evolution equations. Math. Comput. Modell. 57, 1371–1379 (2013)

    Article  MathSciNet  Google Scholar 

  • Shin, B.C., Darvishi, M.T., Barati, A.: Some exact and new solutions of the Nizhnik–Novikov–Vesselov equation using the Exp-function method. Comput. Math. Appl. 58(11), 2147–2151 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  • Sirendaoreji, S.J.: Auxiliary equation method for solving nonlinear partial differential equations. Phys. Lett. A 309, 387–396 (2003)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Taghizadeh, N., Mirzazadeh, M., Rahimian, M., Akbari, M.: Application of the simplest equation method to some time-fractional partial differential equations. Ain Shams Eng. J. 4, 897–902 (2013)

    Article  Google Scholar 

  • Wang, M.L.: Exact solutions for a compound KdV-Burgers equation. Phys. Lett. A 213, 279–287 (1996)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Weiss, J., Tabor, M., Carnevale, G.: The Painleve property and a partial differential equations with an essential singularity. Phys. Lett. A 27, 205–208 (1985)

    Google Scholar 

  • Wen, X.Y.: Construction of new exact rational form non-travelling wave solutions to the (2+1)-dimensional generalized Broer–Kaup system. Appl. Math. Comput. 217, 1367–1375 (2010)

    MathSciNet  MATH  Google Scholar 

  • Xie, F., Zhang, Y., Zhuosheng, L.: Symbolic computation in non-linear evolution equation: application to (3+1)-dimensional Kadomtsev–Petviashvili equation. Chaos Solitons Fractals 24, 257–263 (2005)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Xin, Z., Deng-Shan, W.: A generalized extended rational expansion method and its application to (1+1)-dimensional dispersive long wave equation. Appl. Math. Comput. 212, 296–304 (2009)

    MathSciNet  MATH  Google Scholar 

  • Yomba, E.: The extended fan sub-equation method and its application to the (2+1)-dimensional dispersive long wave and Whitham–Broer–Kaup equations. Chin. J. Phys. 43(4), 789–805 (2005)

    MathSciNet  Google Scholar 

  • Yomba, E.: The modified extended Fan sub-equation method and its application to the (2+1)-dimensional Broer–Kaup–Kupershmidt equation. Chaos Solitons Fractals 27, 187–196 (2006)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Younis, M., Rehman, H., Iftikhar, M.: Computational examples of a class of fractional order nonlinear evolution equations using modified extended direct algebraic method. J. Comput. Methods Sci. Eng. 15, 359–365 (2015)

    MathSciNet  MATH  Google Scholar 

  • Zheng, C.L., Fang, J.P.: New exact solutions and fractal patterns of generalized Broer–Kaup system via a mapping approach. Chaos Soliton Fract. 27, 1321–1327 (2006)

    Article  ADS  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, 430074, People’s Republic of China

    Kalim U. Tariq

  2. Department of Mathematics, Mirpur University of Science and Technology, Mirpur (AJK), 10250, Pakistan

    Kalim U. Tariq

  3. Mathematics Department, Faculty of science, Taibah University, Al-Madinah Al-Munawarah, Saudi Arabia

    Aly R. Seadawy

  4. Mathematics Department, Faculty of Science, Beni-Suef University, Beni Suef, Egypt

    Aly R. Seadawy

  5. Centre for Undergraduate Studies, University of the Punjab, Lahore, 54590, Pakistan

    Muhammad Younis

  6. Department of Mathematics, COMSATS Institute of Information Technology, Lahore, Pakistan

    S. T. R. Rizvi

Authors
  1. Kalim U. Tariq
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Aly R. Seadawy
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Muhammad Younis
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. S. T. R. Rizvi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Aly R. Seadawy.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Tariq, K.U., Seadawy, A.R., Younis, M. et al. Dispersive traveling wave solutions to the space–time fractional equal-width dynamical equation and its applications. Opt Quant Electron 50, 147 (2018). https://doi.org/10.1007/s11082-018-1400-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11082-018-1400-9

Keywords

Search

Navigation